I was not thinking of the "arrow of time"; I was just trying to think of early math techniques that approached (asymptotically?) the concept of "number line". Hipparchus and his Stereographic Projection (not "Stereographic Transformation" as I incorrectly wrote) came to mind for the irrelevant reason that my daughter once used it to solve a homework problem ("show a one-to-one correspondence between the interval (0,1) and the real number line").

As you have pointed out, the Egyptians had an elaborate system of rational numbers. Did they know of the existence of irrational numbers? Or did the Pythagoreans make that discovery? I honestly don't know.

Thank you for discussing the 'arrow of time'. It can be offered as a central mathematical arithmetic metaphor of mathematics. Please continue with the discussion if you so desire.

Historically, well before Dedekind, and modern views of theoretical arithmetic foundations, the Chinese proved the relevance of the circle as a foundation of the Chinese Remainder Theorem, and the eastern form of theoretical arithmetic.

The Greeks followed 1,500 year older Egyptian theoretical arithmetic tradition. Of course, many have disagreed by proposing that Greeks 'invented' the first western theoretical arithmetic.

Today, to avoid the Greek debate, due to the lack of Greek texts, the Babylonian algorithmic metaphor is often used alongside the "Dedekind Cut" to define our modern theoretical base 10 decimal arithmetic.

However, beneath the 'arrow of time', the circle, and other theoretical arithmetic proposals, lie four simple truths. Mathematicians at no time invented the four arithmetic operations. Only applications of the four arithmetic operations have been discussed by Dedekind, the Chinese, Greeks, Egyptians, Babylonians and others.

Best Regards,

Milo Gardner

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